The convergence of the geometric series with r=1/2 and a=1/2
The convergence of the geometric series with r=1/2 and a=1
The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, r and a. The term r is the common ratio, and a is the first term of the series. As an example the geometric series given in the introduction,
may simply be written as
, with and .
The following table shows several geometric series with different common ratios:
Common ratio, r
Start term, a
4 + 40 + 400 + 4000 + 40,000 + ···
9 + 3 + 1 + 1/3 + 1/9 + ···
7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···
3 + 3 + 3 + 3 + 3 + ···
1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···
3 − 3 + 3 − 3 + 3 − ···
The behavior of the terms depends on the common ratio r:
If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where r is one half, the series has the sum one.
If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.)
If r is equal to one, all of the terms of the series are the same. The series diverges.
If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.
The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely-many terms. The sum can be computed using the self-similarity of the series.
Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.
Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.
Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.
Assuming that the blue triangle has area 1, the total area is an infinite sum:
The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives
This is a geometric series with common ratio 1/4 and the fractional part is equal to
For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is
The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is
Thus the Koch snowflake has 8/5 of the area of the base triangle.
The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with .
For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + ), where is the yearly interest rate.
Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + )2 (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is
which is the infinite series:
This is a geometric series with common ratio 1 / (1 + ). The sum is the first term divided by (one minus the common ratio):
For example, if the yearly interest rate is 10% ( = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.